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A (real or complex) Banach space E is said to have the unconditionaly property for martingale differences (UMD-property, for short) if E-values martingale differences are unconditional in L(E;[0,1]). The main reason for the interest in this new class of spaces is that the analogues of several classical results on martingales and singular integrals are also true for a Banach space belonging to this class.
The behavior of compactness under real interpolation real is discussed. Classical results due to Krasnoselskii, Lions-Peetre, Persson, and Hayakawa are described, as well as others obtained very recently by Edmunds, Potter, Fernández, and the author.
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